# Neighborhood

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

A set N is a neighborhood of x exactly when N contains an open set containing x; N itself need not be open.

mko-neighborhood · definition · v0.10.0



## Mathematical statement

In a topological space (X,τ), N⊆X is a neighborhood of x∈X exactly when there is U∈τ with x∈U⊆N. The open sets containing x form an open neighborhood base, but a general neighborhood need not itself be open.





## Representative formula (exact TeX)

x\in N





## Assumptions


- X, τ, x and candidate N are explicit, with x∈X and N⊆X.
- Neighborhoods use containment of an open set; no distance, radius or single least neighborhood is assumed.





## Symbol scope


- x — specified point
- N — candidate neighborhood





## Intuition, definition, and misconceptions

To test N, search all open witnesses containing x and contained in N. The candidate itself can be nonopen and still be a neighborhood.

Neighborhoods are upward closed: if N is a neighborhood of x and N⊆M⊆X, then M is also a neighborhood of x.

A finite topology permits a complete inventory of a point’s neighborhoods. This inventory depends on both topology and point, not just the underlying set.

Do not identify neighborhoods with open sets or open balls, or infer the neighborhood condition from x∈N alone.





## Adopted derivation guide

[Neighborhood: searching for open witnesses and listing all neighborhoods in a four-point topology](https://ocme.evemisslab.com/en/derivations/derivation-mko-neighborhood-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Topological space](https://ocme.evemisslab.com/en/mko/mko-topological-space/) — Neighborhoods are derived from the open family.
- [Open and closed sets](https://ocme.evemisslab.com/en/mko/mko-open-closed-set/) — The witness U must be open.
- [Set membership](https://ocme.evemisslab.com/en/mko/mko-set-membership/) — The point must belong to the open witness and candidate neighborhood.





## Verification boundary

Complete finite powerset enumeration checks the specified model; no metric-ball or general filter theorem is inferred.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.





## Next obligation

Bind nhds, nhds_basis_opens and the upward-closure semantics of neighborhood filters.





## Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.

[Canonical MKO JSON](https://ocme.evemisslab.com/data/mko/neighborhood.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-neighborhood/index.md)